Migration velocity analysis using limited-aperture and monte carlo migration

ABSTRACT

Disclosed is a method for efficiently and accurately determining subsurface velocities for use in migration of seismic data. The method calls for restricting the number of traces considered to those lying upon that portion of the Kirchhoff summation curve wherein the integrand for Kirchhoff migration is smooth. In the preferred embodiment, only a random sample of traces within this aperture are used in the calculations. Improvements in efficiency on the order of a factor of 1000 can be realized with the preferred embodiment.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to the field of seismic imaging, particularly totwo-dimensional and three-dimensional migration of surface seismic data.It consists of a method for efficiently obtaining an accurate velocitymodel, which model can be used in migration of seismic data.

2. Related Art

In seismic exploration of the earth, seismic energy is imparted to theearth. This energy travels into the earth and is reflected by theinterfaces (reflectors; as used herein, "reflector" means the actualsubsurface location of an interface and "reflection" means its apparentlocation by reference to unmigrated seismic data) between varioussubsurface formations. In typical seismic exploration, energy isimparted to the earth at a shot point location, and recorded at a numberof geophone locations spaced at various distances away from the shotpoint. These various distances are termed "offsets." The offsets rangetypically on the order of 50 to 20,000 feet away from the shot point.The shot point is relocated, or a plurality of shot points are used, soas to obtain a plurality of traces at each geophone.

The output signal from each of the geophones is recorded as a functionof time. It is desirable to convert this information so that the picturegenerated by displaying the traces actually corresponds to the depth ofthe various reflectors within the earth. In order to be able totranslate this data into amplitude versus depth, rather than amplitudeversus time, information, the velocities of the various subsurfaceformations must be determined. Accordingly, in order to provide accuratepictures of the subterranean structure of the earth, improved methods ofdetermining the correct velocity of the seismic energy in thesubterranean formations are required. These methods generally areperformed on computers, particularly supercomputers able to manipulatelarge amounts of data efficiently.

The term "migration" refers to correction of data which were recorded asa function of time for the velocity of the wave in the subterraneanstructure. In this process, one can convert a number of offset versustime records, which records can then be displayed to yield a realisticpicture of the structure.

Seismic migration requires an accurate model of the subsurface velocity.There are many existing methods for performing migration velocityanalysis. Three are of particular significance: iterative prestackmigration, prestack migration velocity sweeps and depth focusinganalysis. The methods discussed herein all involve iterative prestackmigration of the data with different velocities, to obtain anapproximation of the velocity by trial and error.

One method for obtaining migration velocities is to prestack migratesubsets (usually common-shot gathers or common offset gathers) of theseismic data with an initial reference migration velocity (Al-Yahya, K.M., "Velocity Analysis by Iterative Profile Migration," Geophysics,54(6):718-729 (1989); Deregowski, S. M., "Common-Offset Migrations andVelocity Analysis," First Break, 8(6): 224-234 (1990)). If thismigration produces images that are consistent for all the data subsets,then the initial guess for the reference velocity is taken to becorrect. If this initial migration produces inconsistent images, thenthese differences can be used to estimate a corrected velocity that iscloser to the true velocity than the initial velocity selected. A flowchart for this method is set forth in FIG. 1. It can be seen from thischart that the calculations are entirely sequential, with the resultthat this method takes not only significant computer time, but alsosignificant interpretation time. This method usually requires severaliterations using the updated velocity for prestack migration each time.Unfortunately, this method is very expensive since prestack migrationitself is very expensive. In particular, the cost of CPU time in asupercomputer to output a single three-dimensional prestack migratedline is presently on the order of $200,000.

The prestack migration velocity sweep method is closely related to theiterative profile migration method, but performs multiple migrations inparallel rather than sequentially. A migration velocity sweep consistsof prestack migrating common-offset gathers simultaneously with severaldifferent velocities and summing the migrated images. This produces aset of seismic traces, one group of traces for each velocity, that canbe plotted to form a velocity analysis display (see FIG. 2 for flowchartof this method). Velocities that produce consistent images with respectto the different common-offset gathers will produce amplitude peaks onthis velocity analysis display. Thus, amplitude peaks on the velocityanalysis display can be used to pick the migration velocity function.Because this method involves parallel processing, less interpretationtime, but more CPU time, is used than with the iterative prestackmigration method.

One problem with this prestack migration velocity sweep velocityanalysis method is that amplitude peaks can appear at velocities that donot correspond to consistent imaging of the common-offset gathers. Thiscan lead the interpreter to pick incorrect velocities (Schleicher, K.L., Grygier, D. J., et al., Ed., Migration Velocity Analysis: AComparison of Two Approaches, 61st Annual Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, Tulsa, Soc. Expl. Geophys., 1237-1238(1991)). These erroneous velocity estimates are the result of dippingreflectors migrating into the velocity analysis location as themigration velocity changes (see FIGS. 3A and 3B). FIGS. 3A and 3B areschematic diagrams illustrating how prestack migration velocity sweepscan produce false velocity picks 6 for dipping reflectors. For thefaulted reflector 2 shown therein, notice that the event has notmigrated into the velocity analysis location 4 at velocity V₁ (FIG. 3A)but at velocity V₂ (FIG. 3B) the event has migrated into the velocityanalysis location. Thus, there will be a relatively high amplitude atvelocity V₂, even though velocity V₁ may be the velocity which producesthe most consistent image as a function of source-receiver offset.

This prestack migration velocity sweep method is about ten times morecomputer intensive than iterative prestack migration, because itrequires 10 to 50 applications of prestack migration. However, thesecomputer costs may be offset by the reduced interpretation costs, sincethe velocity interpreter need pick the migration velocity only once.

A third method for velocity analysis is depth focusing analysis. Depthfocusing analysis determines velocities by using downward extrapolationto estimate zero-offset seismic traces at a range of depths. Theseextrapolations to different depths are all performed with one referencemigration velocity. If a reflection has an amplitude peak at the depthcorresponding to the two-way vertical traveltime through the referencevelocity field, then the reference velocity is the correct migrationvelocity. Deviations from this condition can be used to estimate theerror in the reference velocity (Kim, Y. C. and Gonzalez, R., "Migrationvelocity analysis with the Kirchhoff Integral," Geophysics 56(3):365-370 (1991); Yilmaz, O. and Chambers, R. E., "Migration VelocityAnalysis by Wave-Field Exploration," Geophysics, 49(10):1664-1674(1984)). This method is usually iterated several times until convergenceis achieved.

This method can produce false velocity picks 6 for dipping reflectors,which false picks 6 are similar to those produced by prestack migrationvelocity sweeps discussed above and shown in FIG. 3. (MacKay, S., andAbma, R., Ed., Depth Focusing Analysis Using a Wavefront-CurvatureCriterion, 62nd Annual Internat. Mtg., Soc. Expl. Geophys., ExpandedAbstracts, Tulsa, Soc. Expl. Geophys, 927 (1992)). Furthermore, thismethod may not converge for steeply dipping reflectors (MacKay, S. andAbma, R., "Imaging and Velocity Estimation with Depth-FocusingAnalysis," Geophysics, 57(12): 1608-1622 1992).

All the velocity analysis methods discussed above require severalapplications of prestack migration. Therefore, reducing the cost ofprestack migration would have a significant positive impact on the costof any of these velocity analysis methods.

One way of reducing the cost of prestack migration is to use aninherently fast technique for migration. There are three commonly usedwave-equation migration algorithms: frequency-wavenumber migration,finite-difference migration, and Kirchhoff migration. Frequencywavenumber migration and finite-difference migration are generallyfaster than Kirchhoff migration; however, Kirchhoff migration hasseveral advantages that have made it the method of choice forthree-dimensional prestack migration.

First, Kirchhoff migration can handle irregular shooting geometries,such as those commonly encountered in unstacked three-dimensional data.Second, complex migration velocity fields can be used with Kirchhoffmigration. Third, Kirchhoff methods can migrate reflectors having verysteep dip. Finally, Kirchhoff migration can be used in a target-orientedmode. In this mode, images at a few selected target locations can beproduced at a fraction of the cost of using Kirchhoff migration toproduce images at all possible output locations. Frequency-wavenumbermigration fails with irregular shooting geometries or complex migrationfields, while finite-difference migration fails with irregular shootinggeometries or reflectors with a very steep dip. Neitherfrequency-wavenumber nor finite difference migration can be used in atarget-oriented mode. These methods must compute the migrated image atall possible output locations. This is important for migration velocityanalysis, because velocity analysis can usually be performed at a smallfraction of the number of locations at which a seismic image is desired.Therefore, using target-oriented Kirchhoff migration for velocityanalysis can be cost competitive with the inherently fasterfrequency-wavenumber and finite-difference methods. Because of thelimitations of these methods, only methods for speeding up Kirchhoffmigration were explored for the method of this invention.

The migration techniques proposed herein could be used withnon-Kirchhoff migration methods. However, the techniques proposed hereachieve their greatest gain in efficiency when used with migrationmethods that can operate in a target-oriented mode. Therefore, thefrequency-wavenumber and finite difference methods would not achieveefficiency gains as great as would the Kirchhoff method, because theycannot operate in a target-oriented mode. However, there may be othermigration methods, such as Gaussian beam migration (Hill, N. R.,"Gaussian Beam Migration," Geophysics 55(11):1416-1428), that wouldbenefit from incorporating limited aperture migration.

The equations describing Kirchhoff migration are well known in the art(Berryhill, J. R., Ed. Wave Equation Datuming Before Stack, 54th AnnualInternational Mtg. Soc. Expl. Geophys., Expanded Abstracts, Tulsa, Soc.Expl. Geophys., Session:S2.6 (1984); Schneider, W. A., "IntegralFormulation for Migration in Two-Dimensions and Three-Dimensions,"Geophysics 43(1): 49-76 (1978)). Kirchhoff migration involves summingthe input seismic traces along traveltimes corresponding to a pointdiffractor in the subsurface (see FIG. 4). The migration aperture isdefined as all the traces included in this summation for a given outputtrace. The aperture is usually limited to those traces which have bothsource and receiver within a specified distance from the output tracelocation (usually about 5,000 to 25,000 feet).

FIG. 4 is a schematic diagram illustrating the Kirchhoff migrationmethod. Input traces are summed along the Kirchhoff summation curve 8(diffraction traveltime curve) and output at the apex of the curve 12.The aperture 14 contains all traces within a specified distance of theoutput location. For the reflection 10 shown in FIG. 4, only thosetraces within the shaded area 16 contribute significantly to the sum.

Those input traces which contribute significantly have diffractionraypaths that are close to an actual reflection raypath (see FIG. 5).FIG. 5 is an illustration of which input traces will make significantcontribution to the migrated image of a reflector 17. Sources areindicated by 19 and receivers are indicated by 21. Notice that thetraces that contribute significantly 22 will have diffraction raypathsto the imaging point that are close to a reflection raypath 20 for thatpoint, while those traces 18 which do not contribute significantly havediffraction raypaths to the imaging point that are not close to areflection raypath 20 for that point. Thus, given knowledge of thereflection raypaths 20, raytracing can be used to determine which traces22 will contribute significantly to the output migrated trace.Raytracing is a technique familiar to those of reasonable skill in theart. Including only those traces that contribute significantly in theKirchhoff summation speeds up two-dimensional prestack migration byabout a factor of 10 and speeds up three-dimensional prestack migrationby about a factor of 100.

Just such a method has been developed by Carroll et. al. (Carroll, R.J., Hubbard, L. M., et al., "A Directed-Aperture Kirchhoff Migration,"Geophysical Imaging, Symposium of Geophysical Society of Tulsa, Tulsa,Soc. Expl. Geophys., 151-165 (1987)). They have developed a method forreducing the cost of Kirchhoff prestack migration. They first make areflector model, based on stacked seismic data. Raytracing is used todetermine the locations of sources and receivers that will contributesignificantly to the prestack migration of each reflector. They thendefine a time-varying aperture for prestack migration, centered on theseraytraced locations, that is, significantly narrower than a conventionalmigration aperture (see FIG. 6). FIG. 6 is a schematic diagramillustrating Carroll et. al.'s directed-aperture migration method.Migration hyperbolas are indicated by 28. Normal incidence ray tracingis used to determine the directed aperture 26 used to produce a migratedtrace at the output location 30. Since computer CPU time for Kirchhoffmigration is proportional to the aperture size, this reduction inaperture should greatly reduce the cost of prestack migration. Carrollet al. call this method directed-aperture migration, since the locationof the aperture is moved to different locations depending on a model ofthe reflectors 24.

The method of Carroll et. al. should significantly reduce the CPU timefor prestack migration; however, their method presents a problem. Themethod still requires reading a large fraction of the input traces toproduce a migrated output trace at one location. The reason for this isthat the Carroll et al. aperture varies with time; different sets ofinput traces contribute to the output trace at different times (see FIG.6). Thus, even though only a small percentage of the input tracescontribute at any particular time on the output trace, a much largerpercentage of the input traces are required to form all the time samplesof an output trace. This problem intuitively means that computer I/Ocosts will probably not be significantly reduced, even though the CPUcosts should be reduced.

It is an object of this invention to present a method for performingprestack migration at a dramatically reduced cost.

It is a further object of this invention to present a method forcalculating subsurface velocities at a dramatically reduced cost.

It is a further object of this invention to present a method forcalculating subsurface velocities more quickly and more accurately thanis possible using current methods.

It is a further object of this invention to eliminate false velocitypicks in the determination of subsurface velocities.

It is a further object of this invention to present a method forbuilding an accurate three-dimensional migration velocity from a grid oftwo-dimensional lines.

It is a further object of this invention to present a method foraccurately migrating an existing three-dimensional stack withoutincurring the large expense of obtaining and reprocessing thethree-dimensional unstacked tapes.

It is a further object of this invention to significantly improve thesignal-to-noise ratio (S/N) of velocity analysis displays.

Further objects and advantages of this invention will be seen by oneskilled in the art of geophysical data processing upon review of thespecification, figures and claims herein.

SUMMARY OF THE INVENTION

This invention is a method for analyzing seismic signals for analysis ofa subsurface volume of the earth, comprising the steps of

a. collecting a set of seismic traces;

b. selecting a surface location;

c. determining the migrated positions of reflection dips, whichreflection dips before migration appear to be below the location;

d. selecting a zone on the surface around said location, which zone isat most half as large as the migration aperture required to image allpoints located vertically beneath said location;

e. selecting those traces which have source-receiver midpoints fallingwithin said zone; and

f. performing migration velocity analysis on said selected traces at themigrated position of each reflection.

This invention reduces the cost of migration velocity analysis byreducing the amount of computer time required for prestack migration.Two methods are used to reduce this computer time. Both methods increasethe efficiency of Kirchhoff migration by limiting the amount of inputseismic data.

The first method, limited-aperture migration, is the method set forthabove. It reduces the size of the migration aperture.

The second method, Monte Carlo migration, is used in conjunction withthe limited aperture migration in the preferred embodiment of thisinvention. Monte Carlo migration increases efficiency by migrating onlya small, randomly selected fraction of the input traces. These methodssignificantly reduce the cost of prestack migration and eliminatespurious velocity picks from velocity analysis displays. In addition,using the limited aperture method without the Monte Carlo techniqueimproves S/N over conventional techniques.

Both methods can be applied to either two-dimensional orthree-dimensional seismic data. However, for three-dimensional data thegain in efficiency is about a factor of 10 larger than fortwo-dimensional.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a flow chart for a typical iterative profile migrationvelocity analysis.

FIG. 2 depicts a flowchart for a typical prestack migration velocitysweep method.

FIGS. 3A and 3B are schematic diagrams illustrating how prestackmigration velocity sweeps can produce false velocity picks for dippingreflectors. FIG. 3A depicts migration at velocity V₁, while FIG. 3Bdepicts migration at velocity V₂.

FIG. 4 is a schematic diagram illustrating the Kirchhoff migrationmethod.

FIG. 5 is an illustration of which input traces will make significantcontribution to the migrated image of a reflector.

FIG. 6 is a schematic diagram illustrating the directed-aperture methodof Carroll et al.

FIG. 7 is a schematic diagram illustrating the limited-aperturemigration method of this invention.

FIG. 8 is a flowchart for the preferred embodiment of thelimited-aperture common-offset migration velocity analysis method ofthis invention.

FIG. 9 shows processes used to quantify residual moveout and update themigration velocity for common-offset migration.

FIG. 10 is a velocity analysis display of the data in the Example,resulting from the first iteration of a conventional wide-aperturemigration.

FIG. 11 is a velocity analysis display of the data in the Example,resulting from the first iteration of limited-aperture migration, themethod of this invention.

FIG. 12 is a velocity analysis display of the data in the Example,resulting from the first iteration of limited-aperture/Monte Carlomigration, the preferred embodiment of this invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT

The within invention is principally of a limited aperture migrationmethod. In its preferred embodiment, the limited-aperture method is usedin conjunction with Monte Carlo migration, discussed herein.

Conventionally, migration apertures contain all input traces havingsources and receivers within about 5,000 to 25,000 feet of the outputimage location. The limited-aperture portion of this invention migratesrelatively small (500 to 5,000 feet) fixed apertures of the input datato reduce the cost of prestack migration. Since the computer timerequired for Kirchhoff migration is proportional to the amount of inputdata, this results in up to a factor of 10 improvement in efficiency fortwo-dimensional migration. For three-dimensional data, the gain inefficiency is up to a factor of on the order of 100 since the apertureis limited in two directions.

Conventionally, an output image is formed along a straight, verticalline at the center of the migration aperture. However, when migratingsmall fixed apertures, as proposed in this invention, reflections witheven a small amount of dip will migrate out of the region defining theaperture (see FIG. 7). FIG. 7 is a schematic diagram illustrating thelimited-aperture migration method. Reflectors are indicated as 38.Normal incidence raypaths 34 that pass through the center of the inputaperture 32 indicate output locations to which the input aperture makesa significant contribution. These locations define curves 36 and 40along which the images will be computed. Notice that there can be morethan one output image curve.

Examination of FIG. 7 makes it clear that images cannot, in limitedaperture migration, be constructed along straight, vertical lines. Thekey to limited aperture migration is that images are constructed at thelocations 36 and 40 to which the reflections, at the center of thelimited aperture 32, will migrate. These locations describe curves 36and 40 (see FIG. 7) that replace the vertical lines conventionally usedto form migrated images.

In order to gain a computational advantage from limited-aperturemigration there must be an inexpensive method for determining the outputimage curves 36 and 40. As is known to those of reasonable skill in theart, there are many methods for defining these curves 36 and 40. Onegood method is to digitize reflections on an existing stack of theseismic data. The time dips of these digitized reflections are thencalculated at the center of the limited aperture 32 to be migrated.Kinematic migration (often called map migration (Maher, S. M., andHadley, D. M., Ed., Development of an Accurate, Staple and InteractiveMap Migration Algorithm, 55th Annual Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, Tulsa, Soc. Expl. Geophys., Session:S15.8(1985)) is used to predict, from these time dips, the locations to whichthe reflections will migrate. This kinematic migration is performed withthe same reference migration velocity that will be used for prestackmigration velocity analysis. Output image curves 36 and 40 are thendefined which pass through the output locations 36 and 40 predicted bykinematic migration (see FIG. 7). All the processes used to determinethese output curves are much less expensive than prestack migration.

The output curves 36 and 40 are defined using stacked data, and aretherefore guaranteed to be correct only for small source-receiver offsetinput traces. For steeper reflector dips, large source-receiver offsetswill migrate to different locations than small offsets. This problem canbe overcome by increasing the size of the limited aperture 32. As wouldbe apparent to one of reasonable skill in the art, the proper size ofthe limited aperture 32 is data-dependent, and many methods could beused to determine the proper size of the limited aperture 32. One methodis to run limited-aperture migration for several different aperturesizes at the location in the seismic data containing the steepest dips.These aperture size tests should be performed at a relatively highreference velocity to produce a conservative estimate for the minimumaperture size. The minimum test aperture size that still produces a goodmigration can be used to migrate the remainder of the data. Moresophisticated tests, as would be apparent to one of reasonable skill inthe art, for instance using ray tracing, could be used to determine anoptimum aperture size at each location in the seismic survey.

For three-dimensional seismic data it may be advantageous to use anaperture 32 that has different sizes in the inline and crosslinedirection. In particular, for marine three-dimensional data the inlineprojection of the source-receiver offset usually has a much larger rangethan the crossline projection. Thus, in the crossline direction the dataappears to be essentially zero offset. This implies that the limitedaperture 32 can be much smaller in the crossline direction than theinline direction. Land three-dimensional data, gathered using a swathtechnique, can also benefit by using smaller crossline apertures thaninline apertures.

Two-dimensional seismic data are a limiting case having an aperturewidth in the crossline direction that is essentially zero. We have foundthat the techniques employed for limited aperture migration apply evenin this limiting case. Velocity analysis displays resulting fromapplication of limited aperture migration to two dimensional data are asaccurate as those obtained from three dimensional data, though thedisplays are noisier. This implies that accurate three dimensionalvelocities can be obtained from a grid of two dimensional seismic lines.In this case the three dimensional dips, required for limited aperturemigration, can be determined at the intersections of the two dimensionallines in the grid or from a coincident three dimensional survey. Thiscapability is important, because it can be used to determine accuratethree dimensional velocities without incurring the large expense ofpurchasing an unstacked three dimensional seismic survey. Suchvelocities would be useful, for example, for post-stack migration of astacked three dimensional survey or for map migration of a grid of twodimensional lines.

Limited-aperture migration produces migrated seismic data with higherS/N than conventional migration (Carroll, Hubbard et al. 1987; Krebs, J.R., "Three-Dimensional Migration of Swath Surveys," Geophysics,55(9):1251-1259 (1990)). The reason for this is that limited-aperturemigration sums only over those portions of the Kirchhoff summation curve8 that make a significant contribution 16 to the migration of thereflection of interest 10 (see FIG. 4). The remaining portions of thesummation curve 8 usually do not sum to zero. This non-zero sum is noisethat conventional wide-aperture migration adds to the image of thereflection.

In the preferred embodiment of the method of this invention, Monte Carlomigration is used in conjunction with limited-aperture migration. MonteCarlo migration is the application of Monte Carlo integration theory tomigration of seismic data.

Monte Carlo integration is a well known mathematical technique forestimating the value of a multidimensional integral having a smoothlyvarying integrand within the region to be integrated (Press, W. H.,Flannery, B. P., et al., Numerical Recipes: The Art of ScientificComputing, Cambridge, Cambridge University Press, pages 126-130 (1986)).Rather than summing the function to be integrated over a uniformlysampled region, Monte Carlo integration sums over a sparse randomsampling of the region. This can greatly reduce the amount of computertime required to compute an integral.

Kirchhoff prestack migration is a multi-dimensional integral, and theMonte Carlo method can be applied by simply rejecting, preferablyrandomly rejecting, some percentage of the input traces beforemigration. Unfortunately, when this is applied to conventionalwide-aperture Kirchhoff migration, the result is unacceptably noisy. Thecause of this noise is that the integrand for Kirchhoff migration is notsmooth over the entire aperture, and therefore violates an assumption ofthe Monte Carlo method. In particular, the integrand for Kirchhoffmigration is smooth only over those portions of the aperture thatcontribute significantly to the output image. Therefore, any migrationtechnique which limits the region of integration to this smooth portionof the integrand can be enhanced by use of Monte Carlo migration. Infact, the Monte Carlo method could be combined with thedirected-aperture method of Carroll et al. to further improve thatmethod. Since limited-aperture migration limits the region ofintegration to this smooth portion of the integrand, Monte Carlotechniques can be used in conjunction with limited-aperture migration,even though they would fail with conventional Kirchhoff migration. Aswill be seen herein, Monte Carlo migration does reduce S/N even whencombined with limited-aperture migration. Good results have beenachieved with only a random 10 percent of the input traces being used.Further, even when 90 percent of the input traces are randomlyeliminated from migration velocity analysis, combinedlimited-aperture/Monte Carlo migration still achieves a S/N that isbetter than that achieved by conventional prestack migration. Thisresults in an additional factor of 10 reduction in computer costs. Thus,if the improved S/N afforded by limited-aperture migration is notimportant for a particular set of data, the Monte Carlo method can beused to reduce the cost of velocity analysis beyond that achieved byusing limited-aperture migration alone. The net result is S/Napproximately identical to that achieved by conventional migrationmethods at significantly less than 1% of the cost of computer time.

The Monte Carlo method could be particularly important when usingiterative velocity analysis methods. Iterative methods require readingthe unstacked seismic data many times. Unstacked three-dimensional datacan contain on the order of 1,000 tapes; thus, even the simple act ofreading all of these tapes several times can lead to considerableexpense. However, when using the Monte Carlo method, approximately 90percent of the input traces are randomly rejected before velocityanalysis begins. Thus, each iteration of velocity analysis will requirereading only about 10 percent of the number of tapes that wouldotherwise be required.

Limited-aperture and Monte Carlo migration can be used to improve anyvelocity analysis method which uses prestack migration or wave equationextrapolation. FIG. 8 is a flowchart showing how these methods would beused with iterative profile prestack migration. The flow chart is verysimilar to that shown in FIG. 1, except some premigration work must bedone to determine the output curves for each limited aperture, and,further, the optional Monte Carlo restriction is indicated on FIG. 8.

The following is a summary of the advantages gained by this invention:

1. Reduced CPU and I/O time for migration velocity analysis.Limited-aperture migration velocity analysis for two-dimensional datareduces CPU and I/O times by about a factor of 10, and forthree-dimensional data the reduction is about a factor of 100. By usingMonte Carlo migration another factor of 10 reduction can be achieved,for a total reduction of 1,000 times for three-dimensional data.

2. Improved S/N of velocity analysis displays.

3. With the limited aperture method, two-dimensional lines can betreated as if they were three-dimensional data with a very narrowlimited aperture in the crossline direction. Thus, a grid oftwo-dimensional lines can be used to determined velocities that areaccurate for three-dimensional migration. This could result in a verylarge cost savings for three-dimensional velocity analysis, sincevelocities for three-dimensional post stack migration can be producedwithout purchasing three-dimensional unstacked tapes.

4. The fixed aperture for this invention forces the use of differentoutput curves for each velocity in a prestack migration velocity sweep.This has the benefit of eliminating the false velocity picks frommigration velocity sweep displays discussed above.

5. The number of tapes read in each iteration of velocity analysis canbe reduced by about a factor of 10 by using the Monte Carlo migrationmethod.

Carroll et. al.'s directed-aperture migration and the limited-aperturemigration method proposed here should produce similar gains in CPU timeefficiency. Also, the S/N improvement of migrations produced by the twomethods should be similar. However, there are several major differencesbetween this invention and Carroll et. al.'s.

The main difference is that this method uses a fixed aperture of inputdata while calculating images along output curves that are not vertical.Carroll et. al.'s directed-aperture technique computes a time varyingaperture while calculating images along vertical lines. This differenceresults in the following advantages that were discussed above and cannotbe achieved with the Carroll et. al.'s method:

1. Improved I/O performance.

2. Determination of accurate three-dimensional migration velocities froma grid of two-dimensional lines.

3. Elimination of false velocity picks from velocity sweep displays.

4. Reduced number of tapes read per iteration of velocity analysis.

Note that Carroll et. al. do not propose using their method formigration velocity analysis. Rather, they use their method only to speedup the final prestack migration after another method was used todetermine the migration velocities. Also, Carroll et. al. do not useMonte Carlo migration to produce an additional factor of 10 improvementin efficiency.

EXAMPLE

The following is an example of limited-aperture migration velocityanalysis using common-offset migration as discussed in the previoussection. The data are from a three-dimensional marine survey. The datawere processed as shown in FIG. 8. To quantify the residual moveout andcalculate an updated migration velocity, the fairly simple process shownin FIG. 9 was applied. This method converged after four iterations ofprestack common-offset migration. Other methods of quantifying moveoutand updating the velocity could be substituted, and may lead to fasterconvergence.

The velocity analysis was performed using conventional wide-apertureprestack migration (aperture 26,250×7,875 feet), limited-aperturemigration (aperture 4,375×875 feet), and combined limited-aperture/MonteCarlo migration (90% of input traces rejected for Monte Carlo). Allthree types of migration yielded the same velocity function. Thisagreement confirms the accuracy of the limited-aperture and Monte Carlomigrations.

FIGS. 10-12 compare velocity sweeps resulting from the processes shownin FIG. 9. FIG. 10 shows a velocity analysis display resulting from thefirst iteration of conventional wide-aperture migration. The digitizedcurve is the updated velocity function to be input to the seconditeration of velocity analysis. FIG. 11 shows a velocity analysisdisplay resulting from the first iteration of limited-aperturemigration. The updated velocity curve was digitized from theconventional wide-aperture velocity display shown in FIG. 10. FIG. 12shows a velocity analysis display resulting from the first iteration oflimited-aperture/Monte Carlo migration. Ninety percent of the inputtraces were randomly rejected. The updated velocity curve was digitizedfrom the conventional wide-aperture velocity display shown in FIG. 10.

The accuracy of the limited-aperture and Monte Carlo migration isdemonstrated by the fact that the velocity curve digitized from thewide-aperture migration passes through all the high amplitude velocitypeaks in FIGS. 11 and 12. Also, notice the increased S/N of thelimited-aperture migration relative to the conventional wide-aperturemigration. The Monte Carlo migration also has higher S/N thanconventional wide-aperture migration, though not as good as limitedmigration-aperture alone.

The differences in CPU time for the three methods are shown in Table 1.Limited-aperture migration using the data of this example isapproximately 50 times less expensive than conventional wide-aperturemigration, and Monte Carlo migration gave another factor ofapproximately 10 decrease in expense.

                  TABLE 1                                                         ______________________________________                                        COMPARISON OF CPU TIMES FOR PRESTACK                                          MIGRATION VELOCITY ANALYSIS                                                   Type of Migration    Cray-YMP CPU Time                                        ______________________________________                                        Conventional Wide Aperture                                                                         2,756 seconds                                            Limited Aperture       55 seconds                                             Limited Aperture with Monte Carlo                                                                     6 seconds                                             ______________________________________                                    

What is claimed is:
 1. A method of processing seismic data deriving froma subsurface volume of the earth, comprising the steps of:a) specifyinga family of locations on the surface of said volume; b) for each saidsurface location,i) using reflection time dips to determine subsurfacelocations to which reflections in said data will migrate; ii) using saidsubsurface locations to define a limited migration aperture centered onsaid surface location; and iii) selecting a subset of said seismic data,said data in said subset having midpoints falling within said limitedmigration aperture; and c) performing migration velocity analysis oneach said subset of said seismic data.
 2. The method of claim 1, whereinsaid reflection time dips are determined from a stack of said data. 3.The method of claim 1, wherein said subsurface locations to which saidreflectors will migrate are determined by kinematic migration.
 4. Themethod of claim 3, wherein output image curves are defined from saidsubsurface locations, said output image curves used to determine saidlimited migration aperture.
 5. The method of claim 1, wherein saidanalysis involves the calculation of an error in a reference migrationvelocity.
 6. The method of claim 1, wherein said analysis involvesiterative prestack migration.
 7. The method of claim 1, furthercomprising the step of eliminating at least a portion of said seismicdata from each said subset, said analysis being performed on said subsetafter said elimination.
 8. The method of claim 7, wherein saidelimination is performed randomly.
 9. The method of claim 7, whereinsaid analysis involves iterative prestack migration.
 10. A method ofprocessing seismic data deriving from a subsurface volume of the earth,comprising the steps of:a) specifying a family of locations on thesurface of said volume; b) for each said surface location,i) determiningtime dips of reflections in a stack of said data; ii) using kinematicmigration to determine, from said time dips, subsurface locations towhich said reflections will migrate; iii) defining output image curvesfrom said subsurface locations; iv) using said output image curves todefine a limited migration aperture centered on said surface location;and iv) selecting a subset of said seismic data, said data in saidsubset having midpoints falling within said limited migration aperture;and c) performing migration velocity analysis on each said subset ofsaid seismic data.
 11. The method of claim 10, wherein said analysisinvolves iterative prestack migration.
 12. The method of claim 10,further comprising the step of eliminating at least a portion of saidseismic data from each said subset, said analysis being performed onsaid subset after said elimination.
 13. The method of claim 12, whereinsaid elimination is performed randomly.
 14. The method of claim 13,wherein said analysis involves iterative prestack migration.
 15. Amethod of processing seismic data deriving from a subsurface volume ofthe earth, comprising the steps of:a) specifying a family of locationson the surface of said volume; b) for each said surface location,i)determining time dips of reflections in a stack of said data; ii) usingkinematic migration to determine, from said time dips, subsurfacelocations to which said reflections will migrate; iii) defining outputimage curves from said subsurface locations; iv) using said output imagecurves to define a limited migration aperture centered on said surfacelocation; and iv) selecting a subset of said seismic data, said data insaid subset having midpoints falling within said limited migrationaperture; and v) performing a Monte Carlo elimination of at least aportion of said seismic data from each said subset; and c) performingmigration velocity analysis on each said subset of said seismic data.16. The method of claim 15, wherein said analysis involves iterativeprestack migration.